Some embodiments of the present disclosure relate to phase modulation and detection in an accelerometer, in which a position of an oscillating spring-mounted mass of the accelerometer is phase modulated to encode position as an instantaneous phase of a carrier wave. In phase modulation, an instantaneous carrier phase changes with the modulating waveform. For example, a sinusoidally-varying voltage V(t) with amplitude V0, carrier frequency ωc which includes phase modulation may be given by the equation:V(t)=V0 cos(ωct+Δϕ)=V0 cos(ωct+m cos ωmt)where ωm is the modulating frequency and m is the modulation index. Using known trigonometric identities, the above equation may be rewritten as:
            V      ⁡              (        t        )                    V      0        =                    cos        ⁡                  (                                    ω              c                        ⁢            t                    )                    ⁢              cos        ⁡                  (                      m            ⁢                                                  ⁢                          cos              ⁡                              (                                                      ω                    m                                    ⁢                  t                                )                                              )                      -                  sin        ⁡                  (                                    ω              c                        ⁢            t                    )                    ⁢              sin        ⁡                  (                      m            ⁢                                                  ⁢                          cos              ⁡                              (                                                      ω                    m                                    ⁢                  t                                )                                              )                    The above expression may further be expanded with Bessel functions:
            cos      ⁡              (                  z          ⁢                                          ⁢          cos          ⁢                                          ⁢          θ                )              =                            J          0                ⁡                  (          z          )                    +              2        ⁢                              ∑                          k              =              1                        ∞                    ⁢                                          ⁢                                    -                              1                k                                      ⁢                                          J                                  2                  ⁢                                                                          ⁢                  k                                            ⁡                              (                z                )                                      ⁢                                                  ⁢                          cos              ⁡                              (                                  2                  ⁢                                                                          ⁢                  k                  ⁢                                                                          ⁢                  θ                                )                                                                    sin      ⁡              (                  z          ⁢                                          ⁢          cos          ⁢                                          ⁢          θ                )              =          2      ⁢                        ∑                      k            =            0                    ∞                ⁢                                  ⁢                              -                          1              k                                ⁢                                    J                                                2                  ⁢                  k                                +                1                                      ⁡                          (              z              )                                ⁢                                          ⁢                      cos            ⁡                          (                                                (                                                            2                      ⁢                      k                                        +                    1                                    )                                ⁢                θ                            )                                          